Inspired by the recent work of [1]}, [2]}, we construct the Gaussian perturbation variable via a continuously parameterized tensorization process. Although the proposed method works in the product probability space as in [3]}, it is monotone path centric, and enables to leverage standard perturbation techniques [4]}, [5]} to prove the optimality of the Gaussian solution. To show flexibility and informatics of our proposed construction, we will recover some existing extremal inequalities under the monotone path arguments, including Liu-Viswanath extremal inequality [4]} and vector generalization of Costa's EPI [7]}.
